The electric properties of single-molecule devices are very sensitive to details of contact formation between the molecule and the metallic electrodes. However the factors that control the electron transfer through the molecule in these devices, corresponding to slightly different molecule-metal attachments, are not well understood. In this work, we used a self-consistent molecular dynamics method to study the effect of symmetric and asymmetric contact realizations on electron transfer between two metallic electrodes through a spatially symmetric conjugated molecule. Our results showed that both symmetric and asymmetric electron transfer, with respect to voltage inversion, can be obtained with the same molecule in agreement with the experiments. Besides, a central factor determining the asymmetric electron transfer through a symmetric molecule, caused by the asymmetric contact realization, is the oscillation of the entire molecule between both electrodes and its distortion.